Finite Group Databasebeta

← #128125

Subgroups — #128125

Order 512.

778 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21740C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#7320C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6#2920—21no—
7C22×#99C412yes—
8C2×#407C411yescharacteristic, complemented
9C22×C22⋊C8C411yescharacteristic, complemented
10C2×#380—21no—
11#1870C412yescomplemented
12#1870C2212yes—
13#1869—21no—
14#1869—21no—
15#1817—21no—
16#1799—21no—
17#1799—21no—
18#1799C412yes—
19#1799C2212yes—
20#1779—21no—
21C2×#342—21no—
22#1466—21no—
23#1466C2212yes—
24#1462—21no—
25#1462C412yescomplemented
26#1460—21no—
27#1460C412yescomplemented
28C23×D4C2×C411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16C2×C412yescomplemented
32C23×C8—21no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4Q811yescharacteristic
44C2×C4×D4C2311yescharacteristic
45C22×C4⋊C4—41no—
46C22×C4⋊C4Q811yescharacteristic
47C22×C22⋊C4—22no—
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4D411yescharacteristic
50C22×C42D411yescharacteristic
51#432—22no—
52#432—41no—
53#431—22no—
54#431—41no—
55#423—41no—
56C2×#99—22no—
57C2×#99—44no—
58C2×#99C2×C414yes—
59#407—22no—
60#407—41no—
61#407—41no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8C2×C412yescomplemented
70C4×M16—41no—
71#380—22no—
72#380—22no—
73#380—41no—
74#380—41no—
75C2×#95—41no—
76#342—22no—
77#342—22no—
78#336—22no—
79#336—22no—
80#323—22no—
81#323—22no—
82#323—22no—
83#323C2×C414yescomplemented
84C25C4211yescharacteristic
85C25C22⋊C411yescharacteristic
86C22×D4—22no—
87C22×D4—22no—
88C22×D4—41no—
89C22×D4—41no—
90C22×D4—42no—
91C22×D4C4211yescharacteristic
92C22×D4C22⋊C411yescharacteristic
93C22×D4C4⋊C411yescharacteristic
94C22×D4C22×C411yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—42no—
104C23×C4—82no—
105C23×C4—82no—
106C23×C4C4211yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C2×M16—42no—
120C2×M16—42no—
121C2×M16—82no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—42no—
125C22×C8—82no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—84no—
147C2×C4⋊C4C4⋊C411yescharacteristic
148C2×C4⋊C4C4⋊C411yescharacteristic
149C2×C4⋊C4C4⋊C411yescharacteristic
150C2×C4⋊C4C2×Q811yescharacteristic
151C2×C22⋊C4—24no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4—44no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4C4211yescharacteristic
160C2×C22⋊C4C22⋊C411yescharacteristic
161C2×C22⋊C4C22⋊C411yescharacteristic
162C2×C22⋊C4C22⋊C411yescharacteristic
163C2×C22⋊C4C4⋊C411yescharacteristic
164C2×C22⋊C4C4⋊C411yescharacteristic
165C2×C22⋊C4C22×C411yescharacteristic
166C2×C22⋊C4C2×D411yescharacteristic
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C4⋊C411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C4⋊C8—42no—
176C4⋊C8—42no—
177#99—44no—
178#99—44no—
179C22⋊C8—24no—
180C22⋊C8—42no—
181C22⋊C8—42no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185C22⋊C8—41no—
186C22⋊C8—42no—
187C22⋊C8—82no—
188C22⋊C8—82no—
189#97—42no—
190C4×C8—42no—
191#95—42no—
192#95—42no—
193C24—21no—
194C24—21no—
195C24—22no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—22no—
201C24—41no—
202C24—42no—
203C24—42no—
204C24—81no—
205C24—81no—
206C24#9511yescharacteristic
207C24#9511yescharacteristic
208C24#9511yescharacteristic
209C24C2×C4211yescharacteristic
210C24C2×C22⋊C411yescharacteristic
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—21no—
215C2×D4—21no—
216C2×D4—22no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—44no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4#10012yes—
228C2×D4#10112yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—44no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—84no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4#9511yescharacteristic
280C22×C4#9511yescharacteristic
281C22×C4#9511yescharacteristic
282C22×C4#9511yescharacteristic
283C22×C4#9511yescharacteristic
284C22×C4#9911yescharacteristic
285C22×C4#9911yescharacteristic
286C22×C4#9911yescharacteristic
287C22×C4#9911yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C4⋊C411yescharacteristic
291C22×C4C2×C4⋊C411yescharacteristic
292C22×C4C2×C4⋊C411yescharacteristic
293M16—84no—
294M16—84no—
295C2×C8—44no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—84no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—84no—
308C4⋊C4—84no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C22⋊C4—84no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—41no—
345C23—42no—
346C23—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—41no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—84no—
364C23—82no—
365C23—82no—
366C23—84no—
367C23—84no—
368C23#34211yescharacteristic
369C23#34211yescharacteristic
370C23#34312yes—
371C23#34412yes—
372C23C2×#9511yescharacteristic
373C23C2×#10011yescharacteristic
374C23C2×#10111yescharacteristic
375D4—42no—
376D4—42no—
377D4—44no—
378D4—44no—
379D4—42no—
380D4—42no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—168no—
432C2×C4#34211yescharacteristic
433C2×C4#34211yescharacteristic
434C2×C4#34312yes—
435C2×C4C2×#9911yescharacteristic
436C2×C4C2×#9911yescharacteristic
437C8—82no—
438C8—84no—
439C8—82no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—21no—
450C22—22no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—41no—
456C22—41no—
457C22—44no—
458C22—44no—
459C22—42no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—81no—
470C22—81no—
471C22—84no—
472C22—81no—
473C22—81no—
474C22—82no—
475C22—82no—
476C22—84no—
477C22#139112yes—
478C22C2×#34211yescharacteristic
479C22C2×#34311yescharacteristic
480C22C2×#34411yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—22no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—164no—
493C4#212112yes—
494C2—22no—
495C2—21no—
496C2—21no—
497C2—41no—
498C2—41no—
499C2—42no—
500C2—41no—
501C2—42no—
502C2—81no—
503C2#810711yescharacteristic
504C2#1257411yescharacteristic
505C2C2×#139111yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC24C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7320C2semidirect
8. C2×#407C4semidirect
9. C22×C22⋊C8C4semidirect
11. #1870C4semidirect
25. #1462C4semidirect
27. #1460C4semidirect
31. C22×M16C2×C4semidirect
69. C2×C22⋊C8C2×C4semidirect
83. #323C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#217402
C2×#17992
#173642
#73202